Wednesday, March 26, 2014

BICEP2 string-theoretical winner: N-flation

Last Tuesday, I discussed some of the winners and losers of the apparent discovery of the primordial gravitational waves.

The first winner that I mentioned was Andrei Linde's "chaotic inflation" with the potential \(V=\frac 12 m^2 \phi^2\). It is extremely simple and it matches all the observations at this point. However, it is just an effective field theory, not a full-fledged compactification of string theory.



What are the actual compactifications or scenarios in string theory that are winners? We will turn to this question later in this blog post.

One point that I haven't sufficiently emphasized is that the anti-physics activists, the sort of computer administrators, senile teaching assistants, and other assorted unfriendly individuals are the top losers – in the excitement following the discovery, we have almost completely forgotten that they exist. Imagine how you must feel if you have been preaching for decades that physics at dozens of \(\TeV\) is untestable, unfalsifiable, or any other adjective that was popular among similar cranks... and suddenly, a cheap $10 million experiment measures the value of a new parameter that describes some physics at the supersymmetric GUT scale, around \(10^{16}\GeV\), and it eliminates 90% of the theories that say something about the scale. 90% is quite a brutal case of falsification, isn't it? It would be appropriate if at least 90% of these physics haters disappeared now, too.

Yakov B. Zeldovich once said that the Universe is the poor man's accelerator. In the USSR, they had sufficiently many poor men which is why they had to look to the Universe and why a significant fraction of the pre-fathers and early co-fathers of inflation were Russians. And as another Slavic physicist and probably the first man who thought about the decomposition of CMB into E-modes and B-modes, Uroš Seljak, observed, "[the discovery] may force us in the direction of string theory; it also fits in with models of continuing inflation that produce multiple universes."




We are suddenly observing physics at time scales of \(10^{-35}\) seconds, length scales \(10^{-27}\) meters, and so on, i.e. billions of times shorter distances and times than those at the LHC, and even though BICEP2 has effectively measured 1-2 new nonzero parameters only, we may proceed with the spring cleaning which eliminates a large portion of the detailed inflationary (and an even higher fraction of alternative!) models. A majority of string compactifications may now be excluded, too.

So which string compactifications and scenarios are dead and which of them look really good right now? The dust hasn't settled yet and there's actually some disagreement. For example, even when we think about string-inspired scenarios in phenomenology, most experts that I communicated with think that the old large dimensions (ADD: Arkani-Hamed, Dimopoulos, Dvali) have been excluded and many of them believe that the warped extra dimensions (RS: Randall-Sundrum) share their fate. However, Lisa Randall thinks that RS is doing fine as the inflation "takes place" at the Planck brane and is unaffected. And Edward Witten was heard as saying that even ADD may actually be viable after BICEP2.




It would be helpful if these physicists revealed some details of their reasoning.

But I think that in general it's true that most "papers on stringy models of inflation" in existing literature focus on models that are now ruled out. While the cyclic and ekpyrotic models predict "virtually zero" B-modes, i.e. the value of the tensor-to-scalar ratio \(r=0\), the inflationary models predict a nonzero \(r\) but in most cases, the predicted value of \(r\) is just much lower than \(r=0.20\pm 0.05\) suggested by BICEP2.

You should understand that \(r=0.20\) is really near the "maximum" value of \(r\) that has been considered in any theoretical paper. Significantly higher values of \(r\) are really of "order one" which seemed "manifestly excluded" to the people which is the probable reason why that option hasn't even been considered. A value of \(r\) of order one would apparently mean that the polarization would have to be "visible to the naked eyes" but it had not been discovered earlier, which is why the people didn't consider it at all. I think that if people wanted, they may easily find models with still larger values of \(r\).

On the other hand, you should understand that values of \(r\) obeying \(r\leq 0.001\) or so are unobservable. It's because the value of \(r\) cannot be measured quite accurately. We measure it from the amplitudes of the spherical harmonics \(Y_{\ell m}\) for \(\ell\approx 100\), and there are just thousands of such particular coefficients. Each of them is inevitably random to some extent and even if you incorporate all of them into your calculation of the "average intensity" or your calculation of \(r\) in order to suppress the noise, some noise just cannot be eliminated. The unavoidable error/noise that results from observing a finite number of coefficients – or a finite number of "vortices in the sky" – is known as the cosmic variance. Cosmic variance is an enemy of the observational cosmologist (yes, the preposterous universe is an enemy, too) because it prevents her from measuring things like \(r\) too accurately. And values \(r\leq 0.001\) or so are simply indistinguishable from \(r=0\).

Many inflationary models – and stringy descriptions of inflation – have predicted much smaller values, \(r\ll 0.001\), so they predicted that the primordial gravitational waves would never be discovered. If BICEP2 is right, these models are excluded.

I also agree with Liam McAllister who thinks that the old large dimensions and similar models are ruled out:
We do learn one model-independent thing about string theory: because the inflationary Hubble scale is so large,\[

H\approx 10^{14}\GeV,

\] we can exclude a wide range of models in which quantum fluctuations at this scale would destabilize the compactification. In particular, if the Kaluza-Klein mass is below \(H\), the same fluctuations that give rise to the scalar and tensor perturbations of the CMB would give rise to perturbations of the extra dimensions. When fluctuations of this sort are large, a four-dimensional description ceases to make sense, because the whole compactification is dynamical. By this logic we can exclude models with very low Kaluza-Klein scales, i.e. models of large extra dimensions. (Perhaps there is a model-building trick that can make large compactification robust against quantum fluctuations during inflation, but I'm not aware of a compelling idea.)
Well, perhaps the whole dynamical compactification could still survive the era of inflation and having a workable four-dimensional effective field theory might be unnecessary but I do agree with him that the default assumption is that a compactification destabilized in this way is probably devastating. I think that this eliminates old large dimensions with radii \(R\gg L_{\rm Planck}\) but also warped geometries with curvature radii or proper lengths of the extra dimensions \(R\gg L_{\rm Planck}\).

Building on the pro-anthropic KKLT paper, six authors KKLMMT 2003 began to study inflation in string theory. I believe that the "D3-brane inflation" described by that paper is excluded by BICEP2 if that experiment is right. It doesn't matter that the paper is approaching 1,000 citations and it doesn't matter that it talks about a formidable number of vacua, like \(10^{500}\); all of them may still be shown wrong. That's the power of the empirical evidence in science! The reason is that the location of the D3-brane in a throat which plays the role of the inflaton has a problem with the Lyth bound that was discussed by Liam: the vev of the field just can't change by higher-than-Planckian values which is needed for significant tensor modes and string vacua without some extra structure ban these large variations of the fields.

We see that while the BICEP2 discovery supports the general concept of the multiverse by strengthening "chaotic inflation" which apparently implies "eternal inflation", it may rule out the single most popular incarnation of the anthropic reasoning within string theory as described in the KKLMMT paper. I have never liked the pro-anthropic KKLT/KKLMMT scenarios and especially the ideology behind them because they would prefer silly arguments based on "majority" and "typicality" and all this left-wing junk as opposed to a clever description of the observed patterns by unique enough physical constructs. So I may be a bit biased against KKLT/KKLMMT. But nevertheless, I do think that this line of reasoning is in trouble. The most unequivocal assertion that these models predict low \(r\) appeared in this Linde-Kallosh 2007 paper.

A string theorist would write me an e-mail considering F-theory to be the main stringy winner of the BICEP2 announcement. Well, some people consider KKLMMT to be a "generic" or "most typical" representation of inflation within F-theory, and it may be ruled out. So the message shouldn't be this clearly positive. I feel that the traditional (and for 20 years, my favorite) heterotic vacua with the Calabi-Yau scale close enough to the Planck scale – and producing effective 4D field theories beneath this high scale – might be doing better.

N-flation: a winner

However, there exists another line of stringy model building that seems to have strengthened substantially. It was started in 2005 and empirically, it is almost exactly equivalent to Andrei Linde's simple "chaotic inflation". The paper was
N-flation by Savas Dimopoulos, Shamit Kachru, John McGreevy, Jay Wacker
and it appeared in JCAP in 2008 (quite a delay). Like the paradigm of the stringy axiverse (which also includes Savas Dimopoulos among the co-authors), N-flation uses the observation that string theory vacua are likely to predict a high number of periodic scalars, the axions. The high number is the reason behind the letter \(N\) in the name – and the name, "enflation", sounds almost like "inflation". The authors must consider this linguistic unintelligibility to be an advantage. ;-)



String theory vacua often predict a whole plethora of different axions. Picture taken from the blog of the UCLA professor who makes sure that Sheldon Cooper's science is kosher.

Note that there is the \(B\)-field in perturbative string theory, \(B_{\mu\nu}\), that plays the role of the electromagnetic potential \(A_{\mu}\) for the "string winding charge". You may integrate it over various 2-cycles of the compactification, and there may be many of them, to produce scalar fields. There may be other differential forms that may be integrated over cycles of appropriate dimensions to yield additional scalars, and so on. All these scalar fields are analogous (and sometimes U-dual to) "angles" of rectangular (I mean paralellogram/general toroidal) compactifications which are periodic variables, too.

If you have many axion fields like that, the potential\[

V = \sum_{i=1}^N \gamma_i a_i^2

\] allows you to pick vevs of the fields, \[

\langle a_i\rangle = \frac{v}{\sqrt{N}},

\] which means that by the Pythagorean theorem, the "collective" overall scalar field \(a\) is of order one (not a decreasing power of \(N\)) even though the individual fields \(a_i\) have vevs much smaller than that, by a factor of \(\sqrt{N}\), so the conflict with the Lyth bound is avoided. So far, I have assumed that the coefficients \(\gamma_i\) were the same for all axions \(a_i\). The authors then refine the model by allowing the axion masses to be different – e.g. uniformly covering the log scale. If that's so, the cosmic inflation naturally has "many stages" in which the "most important" (fastest running) inflaton is gradually switching from the heaviest axion to the lightest axion.

They also discuss various radiative corrections and radiative stability, in the presence of SUSY and without the presence of SUSY, and the important role of the shift symmetry.

The most important point is that this scenario ends up being extremely similar to Linde's simple chaotic inflation – except that it could perhaps also predict some of the non-power-law running that seems to follow from the (so far minor) tension between Planck and BICEP2. In other words, the large and simple form of the tensor perturbations could be evidence for Linde's simplest one-quadratic-scalar model; but it could also be evidence for inflation's dependence on many scalars, i.e. evidence for lots of scalars (axions) in Nature! Somewhat less directly, it could be evidence in favor of a complicated enough compactification of extra dimensions in string theory!

This model of N-flation is a "simpler" and "more stringy" cousin of assisted inflation. Assisted inflation also had the point of showing that inflation may be slow-rolling if there are many scalar fields – even though the rolling would be fast with each scalar field separately. They seem to require a particular form of potential, an exponential one (I really mean \(V=A\exp(B\phi)\)...), which is probably unnecessary. At any rate, I do think that N-flation is smarter, simpler, and contains the main viable ideas that were coined in assisted inflation.

I highlighted N-flation in order to have a particular brand but there are several stringy and semi-stringy models of inflation that are equally capable of predicting large enough tensor-to-scalar ratios. For example, this 2011 paper by Barnaby and Peloso lists several such paradigms:
[...] Moreover, \(f\gt M_p\) does not seem possible in string theory [Banks et al. 2003]. More recently, several controlled realizations of axion inflation have been studied – including double-axion inflation [Kim et al. 2004], N-flation [Dimopoulos et al. 2005, Easther+McAllister 2005], axion monodromy [McAllister et al. 2008, see guest post by Eva Silverstein], and axion/4-form mixing [Kaloper+Sorbo 2009] – which have \(f\lt M_p\) but nevertheless behave effectively as large field inflaton models (\(\phi\gtrsim M_p\)). [...]
Whether these models come in intimidatingly large groups supporting the anthropic delusions or not, they may be picked as the winners by the experiments, and that's what matters more in physics than philosophical preconceptions. I guess that when the dust settles, people will focus much more attention on these scenarios and the experiments may provide us with several new pieces of nontrivial information, too.

Particle accelerators will remain important and complementary but a "new" donor of exciting empirical data, the poor man's accelerator named the Universe, may gain much more importance than what most of us used to expect.

46 comments:

  1. Dear Lubos, "except that it could perhaps also predict some of the non-power-law
    running that seems to follow from the (so far minor) tension between
    Planck and BICEP2. In other words, the large and simple form of the
    tensor perturbations could be evidence for Linde's simplest
    one-quadratic-scalar model; but it could also be evidence for
    inflation's dependence on many scalars, i.e. evidence for lots of
    scalars (axions) in Nature!" ... so, what would be the precision level needed to decide between Linde's and N-flation scenarios? Should we hope to see this in near future? Thanks!

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  2. Hi Lumo,

    I have just started reading this and heartily agree with the paragraphy about computer administrators and teaching assistants. Have you noted that in particular the agressive computar administrator kept silent during all the time since the potential discovery? Of course I dont klick him ;-P, but from the titles in the right side bar of the Uduality blog (I still wonder why Kneemo includes anti-physics blogs there) I have seen that first the Trollking was in "Spring break" and the current title is something so meaningless, that I have completely forgotten it already, LOL :-D. Do you still remember how he celebrated every single null result of experiments looking for new physics by (maybe not chaotically) intlating ;-) it to ridiculous proportions ...?

    And thanks for the nice introductory comments about N-flation, I think the idea that it can be viewed as a generalization fo Linde's chaotic inflation is cute. I will reconsider that part when back home to have some relevant texts I have read within my grasp :-).

    Could you add similar introductary remarks about axion monodromiy inflation too?

    Cheers

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  3. Dear Lubos, I find it interesting that Linde's original paper proposing Chaotic Inflation is so short and (ahem!) simple. Looks like something I might have mastered in my youth in less than ten years! (Probably not so.)

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  4. Dear Numcracker, it's a nice and interesting question and I don't know what is the "canonical" answer to it if there's one. However, one must understand that Linde's model with one scalar is much more unique and robust, so it is easier to falsify it. Multi-inflaton models have many more parameters etc. to adjust (if treated as effective field theories). So every disagreement with Linde's simple theory is an argument in favor of more complex theories. On the contrary, if Linde's models continues to be more or less compatible with everything, particular competing models should be disfavored relatively to the simpler model because they're more complex.


    In principle, I believe that the running spectral index - the main minor "anomaly" observed by the combination of BICEP2 and Planck at this moment - may be explained by a multi-inflaton model (or many other types of more complex models, for that matter) but I haven't really done the calculation to map the parameters.

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  5. Dear Dilaton, very true. Right, the sourball-in-chief had a spring break. From yesterday, he was censoring all comments about BICEP2 and called the result "hype", and I have also seen his blog entry which he tries to turn against string theory. Given the fact that the BICEP2 observation contradicts everything he's been saying about untestability of unifying physics theories etc., I would expect him to splash himself into a toilet. Frankly speaking, I find it totally incredible that he is ready to show his ugly head in the public again, just one week after the announcement.

    The monodromy inflation isn't hard. For a few lines about it (plus the context), see e.g. Matt Reece's post-BICEP notes
    http://users.physics.harvard.edu/~mreece/inflation.pdf

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  6. LOL, the page you linked to is just the abstract. The actual paper has 3 pages


    http://jetpletters.ac.ru/ps/1480/article_22578.pdf



    and yes, it's a great idea for each of us to (try to) read it again at this point.

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  7. Hm, thinking about it, the Trollking is free to behave as the asshole he really is on his own blog, and as the worst asshole that I have ever seen, we should probably not have expected him to react different to the new discovery ... ;-)

    But I expect that his negative impact and influence will diminuish a bit now, as with the current situation even some dimwits will start to see that he is nothing but an aggressive troll who enjoys taking the role of the leading guru of an anti-science sect ...

    Thanks for the link to Matt Reece's notes, this looks nice and I have to print it out :-)

    Cheers

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  8. Sure, he's free. Still, some ethical standards in various environments sometimes influence what and when an individual may do or say. Let's hope you're right and his influence will continue to drop - and that the drop won't be a temporary downward fluke that will be forgotten within a month.

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  9. Lubos,

    Great article! I am slowly starting to appreciate the importance of the large r = .2 value in restricting the possible cosmological models, as your post emphasizes. Once the telescopes reach a resolution pinning down the details on the spectral tilt, do you foresee further restrictions coming on the parameter space or more evidence pointing to stringy models?

    Could you be so kindly as to give a link to some of the details on compactification in string theory? Something on a non-expert level would be perfect, like Preskill’s incredibly masterful “Inflation on the back of an envelope”, but the details on selecting the required homeomorphisms is what really intrigues me.

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  10. Hi Lubos

    I was looking for a paper regarding the implication of SUSY breaking at that scale and today I found this one:

    http://arxiv.org/pdf/1403.6081v1.pdf

    But could you comment on the last paragraph of this paper? I was thinking this on my own too before reading it; it seems that too many scales are squeezed there:-)

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  11. Dear Giotis, thanks for the link to the paper! There are just 3-4 ordered scales in that paragraph. That's nothing remarkable. You can have a hierarchy of 10 or more scales which are order by "<<", much smaller than, and there is often an explanation.


    For example, in perturbative string theory - and it may be a part of the story discussed in the Ibanez et al. paper above - one has the Planck scale which is much higher (energy) than the string scale, and various D-brane scales are in between, and so on.


    It's because the Planck scale is derived from Newton's constant that is, in D=10, g_{closed}^2 * L_{string}^8, so the eighth power of its inverse is g^{-1/4} M_{string}. Various other scales from D-brane tensions etc. have the form g^n * M_{string} with various other numerical values of the exponent "n", and for g<<1, one obviously gets the same "<<" sorting of these scales.


    In physics, we know quite a lot of scales -neutrino mass scales, QCD scale, electroweak scale, superpartner scale, intermediate hidden sector scales, SUSY breaking scale, GUT scale / inflation scale, Planck scale - and they're also ordered by "<<" (much less than). Physics is very diverse when it comes to scales, at least some of these hierarchies are actually needed for life ("anthropically").

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  12. Dear Lubos,


    currently I am looking at the TASI lectures on Inflation and I just noticed that the author works with quantised metric perturbations.



    So am I right that if the result is true, that proves that gravity is quantised? Or is it possible to reproduce the result with classical gravitational waves?

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  13. Dear Rezso, yes, gravity is definitely quantized and the quantized modes of the gravitational field are absolutely needed to explain the observed variations (B-modes) within the inflationary theory. In this sense, the discovery proves that "gravity is quantized".

    I must add two comments - one of them will argue that "gravity is quantized" has always been clear; the other will argue that the proven meaning of "gravity is quantized" remains sort of limited even after the discovery.

    First, we have really known that "gravity is quantized" well before that, at least if we interpret the sentence as "the gravitational force exists in Nature *and* the postulates of quantum mechanics hold in Nature". There's no way how gravity - or any other observable part of reality - could avoid obeying the postulates of QM as well.

    I've discussed this point repeatedly, e.g. here:

    http://motls.blogspot.com/2012/01/why-semiclassical-gravity-isnt-self.html?m=1



    Schrödinger's cat's life may be linked to the fate of a radioactive nucleus. The latter evolves into a superposition of macroscopically distinct states, so must the cat, too. If the cat explodes, its ashes' gravitational field is a bit different. So the gravitational field around the cat - or any other gravitational field - is also generally evolving into superpositions. There is no consistent "mixture" of the laws of classical physics and quantum mechanics, as I try to explain e.g. in the text above.


    Concerning the second point, the nature of "quantum gravity" that is demonstrated by BICEP2 remains "limited" in the sense that all the things that make "quantum gravity hard" may still be avoided in the (accurate enough) calculation of the B-modes. The gravitational field is quantized but the approximation of the "first corrections in the hbar expansion" to the classical equations are enough. One doesn't need to compute any quantum loops (which produce the divergences) to make accurate enough predictions for the B-modes. More precisely, these loop corrections contribute less than 1 percent, and probably much less, even at the "high scale inflation" that seems preferred now.


    So if you don't care about this sub-1-percent accuracy, you may quantize Einstein's equations in the same way as you quantize the electromagnetic field, completely overlook the loop diagrams (that show that quantized GR, unlike QED, is non-renormalizable and sick), and just directly interpret the tree-level calculations from the "naively quantized" Einstein's equations.

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  14. Thanks, I had semiclassical gravity in my mind, where the metric tensor is kept classical and the energy-impulse tensor operator is replaced with it's vacuum expectation value in the Einstein-equations.

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  15. Right, Rezso, this is the kind of approximation that is OK in everyday life because "macroscopically different gravitational fields" don't interfere with each other in practice - we may always imagine that the alternatives decohere and/or are measured before that.


    But fundamentally speaking, this semiclassical gravity is inconsistent, like any mixture of QM and classical physics. The expectation value of the stress-energy tensor in the state "psi" is a nonlinear function of "psi", and if you allowed this expectation value to influence other degrees of freedom described by "psi" (by allowing the expectation value to influence the gravitational field, and by acknowledging that the gravitational fields influence quantum objects in the universe), then you would see that the general "psi" evolves nonlinearly.


    But a nonlinear evolution of "psi" implies either loss of unitarity (the total probability isn't one) or, if you artificially rescale the wave function in some way, loss of locality. Such a nonlinear evolution would really make the question "when did the wave function collapse" physical, and you could use this "now real wave function collapse" to send superluminal signals between entangled pairs, and so on.


    It is absolutely essential that the actual evolution of the state vector is linear, given by a single linear (evolution) operator (or the Hamiltonian). It's needed for the simplicity; it's needed for locality, unitarity, and the avoidance of would-be paradoxes that one could derive from a "genuine wave function's collapse".

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  16. Hi Lubos

    There is later work which show N-flation actually predicts r<<0.1
    http://arxiv.org/pdf/0707.1982v2.pdf
    http://arxiv.org/pdf/1108.2944v1.pdf

    Another string theory based model that claims to give very similar predictions to $m^2\phi^2$ inflation is M-flation
    http://arxiv.org/abs/arXiv:0903.1481
    I would be very interested to hear your opinion of it.

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  17. Dear physicsphile, the M-flation paper is fun (Shahin Sheikh-Jabbari is a co-author - I know him well, we've written something together, too). It contains values for "r" according to the detailed choice in the model: 0.132, 0.26, 0.2, 0.048 (search for "tensor" in that paper), clearly in the right "ballpark".


    The M-flation authors say that their model is a special case of N-flation so if I view them as credible enough, there's evidence that N-flation gives these high values of "r". I will look at the other two papers now (one is newer, one is older than the M-flation paper).

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  18. "a cheap $10 million experiment measures the value of a new parameter" removing defective theory claimed immune to empircal falsification. Inflation's vacuum symmetries can be achiral, racemic, enantiomeric excess, resolved chiral. Our universe is not mirror-symmetric. It contains excess matter angular momentum (Tully-Fisher relation: Milgrom acceleration, arXiv:1310.4009, 0906.0668, versus dark matter curve fit).

    A geometric Eötvös experiment costs ~$0.2 million. If tensors and scalars where pseudotensors and/or pseudoscalars (e.g., false vacuum decay powering inflation), 1) a mirror-asymmetric universe, 2) more matter than antimatter, 3) Milgrom acceleration via Noether and trace vacuum chiral anisiotropy, 4) parity violations, symmetry breakings, chiral anomalies, Chern-Simons repair of Einstein-Hilbert action, 5) string theory contracts to a white-hot needle of inquiry.

    Perform a geometric Eötvös experiment, left-handed versus right-handed (P3(1)21 versus P3(2)21 enantiomorphic space groups ) alpha-quartz as visually and chemically identical, single crystal test masses..

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  19. Hi Lubos, could you elaborate on your comment that N-flation can give some of the "non-power-law running" compared to phi^2. Do you have a reference for this? Thanks

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  20. No, unfortunately, I don't know of a reference. It just seems likely to me that if the inflation is switching from a field to another - in the non-trivial spectrum of the axion masses - and perhaps if the density of the spectrum depends on the mass, one shouldn't get a uniform spectral index.

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  21. OK, thanks.
    By the way, could you also define what is meant by a "string axion"? Why are these fields given the name of "axion"? I am familiar with the QCD axion; so what do these other fields have in common with the QCD axion to deserve the same name?

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  22. Hi, the QCD axion is the scalar-field coefficient of the term "F wedge F" in the QCD Lagrangian. This coefficient is effectively a periodic field because the integral of "F wedge F" is integer (in some units) and the change of the action by a multiple of 2*pi is unphysical.


    The periodicity of the "axions" is what all axions have in common; some shift symmetry "axion goes to axion plus constant" must be unbroken, at least the discrete group. String theory contains lots of fields that are periodic, like angles of the parallelogram (toroidal) compactifications, similar angles, and integrals of differential form fields (potentials) like B_{mn} over 2-dimensional (or similarly higher-dimensional) submanifolds of the manifold of extra dimensions.

    The universal string axion "a" is defined by *da = dB, in terms of differential 3-forms and their Hodge duals, and it plays a special role because it's "independent" of the compactification. The other axions, integrals of the form fields, depend on the compactification, and for each cycle, you get an axion (so their number is given by the Betti numbers etc.).


    String theory axions and QCD axion have something else in common aside from "the defining property", the periodicity. A combination of string axions may physically play the role of the actual QCD axion in the real world. If QCD with the axion is embedded into a complete theory, the QCD axion *has to be* a function/combination of the string axions.

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  23. Hi, thanks for your nice answer. But would it be fair to say that we need more than just periodicity to call a field an axion? For instance, any Goldstone boson should be periodic, e.g., pions should see a periodic potential. But we don't call pions "axions". So what feature do pions lack, that other fields have, that stops us from calling them axions?

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  24. Hi Higgs, this is a neat question but to a large extent, it is a terminological one.

    There are indeed very good reasons to count pions among axions. For example, they have analogous reasons why the potential/masses are small/constrained, and both enter the Primakoff process, see page 15 of these slides

    http://www.physik.uzh.ch/lectures/astro/10/vorlesung/Schumann_Axions.pdf

    or top of page 2 of

    http://arxiv.org/pdf/hep-ph/0611118.pdf

    In this non-stringy context, pions are not called axions because they are not elementary but composite fields.

    In the case of spontaneously broken symmetries, Goldstone bosons are eaten by gauge bosons, so they become new polarizations of the gauge bosons, so their fate is different from proper axions that remain physical scalar particles.

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  25. I naively imagined the "switching to another field" as switching to some kind of another effective regime by some kind of introducing a new (mass) scale which spoils the constancy of the spectral (slope) index ...
    Having many fields with different masses could then in principle spoil the presence of a well defined spectral range with constant spectral slope (index) generally (= running of the spectral index) ...?

    Sorry if this comment is less meaningful than what Uncle Al said ;-), but I could not help giving it a guess ... :-D

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  26. //Also, I was shocked that Russian-language TVs were banned in the whole Ukraine.//


    Well, this is hardly a shocker, given that Russia just took a portion of Ukraine territory by force. It may be stupid, as it could further inflame the Russian speaking residents of Ukraine, but otherwise, it is an understandable reaction.

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  27. Well, it may be understandable, but it is not compatible with standard relationships with a neighbor country where the suppressed minority is the majority.

    In other words, as I already said, it may be understandable, but the Russian occupation of the rest of Ukraine will be equally understandable. To a certain extent, many events that followed the coup in Kiev are inevitable. But responsible people should still try to look ways how to avoid the escalation beyond a certain critical threshold.

    Again, I emphasize that not even Adolf Hitler has ever imposed policies that you find "understandable" in the case of the 21st century Ukraine.

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  28. I must clarify something. I thought you meant they stopped the transmission of TV channels from Russia in Ukraine. If that's what they did, I do find that understandable - who needs propaganda from a country that just grabbed a part of your territory fed to your own population. If they banned local TV stations that transmitted in the Russian language, that is another matter - trying to stamp out the Russian language can only backfire.

    More broadly though, I have been reading your posts on the subject, and I have the impression you are a bit too naive about the purity of Russia's intentions. I am not sure there are any good guys here, but it's certainly not the case that Ukraine is some pure evil while Russia is all innocent. Say what you will, it was an internal problem in Ukraine that triggered Russia's external response. Was the Russian population of Ukraine really threatened?

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  29. The 2007 paper was for quadratic potentials. But they restricted the initial conditions to be within $M_{pl}$ of the origin. More realistically, one needs a periodic potential with a wavelength of less than M_{pl} like in the 2011 paper. In that case, for random initial conditions, in general only one field dominates with the inflaton near the maximum. As it is then effectively a single small field model, r is low.

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  30. Lubos and Rezso,

    What a marvelous discussion on quantum gravity to follow along with. Lubos, your quantum gravity post (linked to) is highly convincing, as is this paragraph above: [The gravitational field is quantized but the approximation of the "first corrections in the hbar expansion" to the classical equations are enough. One doesn't need to compute any quantum loops (which produce the divergences) to make accurate enough predictions for the B-modes. More precisely, these loop corrections contribute less than 1 percent, and probably much less, even at the "high scale inflation" that seems preferred now.]

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  31. Oh yeah, a guest post about monodromy inflation here would be cool :-)))

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  32. The west is in deep shit for arranging this putsch.

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  33. It is quite similar to 1995 Slovak language law (banning in a sense Czech language in Slovakia, main target being Hungarian). Important thing is that Russian and Ukrainian are similar and mutually intelligible. The ban is obviously not meant to cause any real trouble in communication, but just to make people upset and offended.

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  34. " because they would prefer silly arguments based on "majority" and "typicality" and all this left-wing junk" Ha! Yes.

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  35. So a Russian should get the theory price then for b mode.

    Nobody is explaining what happen to the g waves from 1 sec to 370k years. How much were they stretched and attenuated? Phusists can not explain anything properly.

    What was the size of the observable uno before and after.

    Also a quick search shows that the Planck data does not agree.
    Maybe they need the same math trick to pull the rabit out of the hat.

    I hope Planck does not hit them in the face

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  36. Sorry, the behavior of gravitational waves is accurately enough described by the effective field theory (in a curved spacetime background in this case), and it is fully analogous to the description of the electromagnetic fields, e.g. those in the cosmic microwave background.


    There is no significant disagreement between BICEP and Planck, especially if one allows sufficiently general dependence of the amplitudes on "k".


    Moreover, your last "hope" has nothing whatever to do with your previous "complaints". The previous complaints are theoretical in character - but Planck and BICEP are two competing experimental teams that use the same mankind's theoretical knowledge and models of what could have actually happened. So if you believe that Planck will be endorsing your pseudoscientific views about the non-existence of CMB or its polarization or the alleged incompleteness of the theories we use to describe them, you are bound to be disappointed.

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  37. I truly hope you've gone over to the senior teaching assistants blog and read his current posts and the comments. I think you get about a decade of "I told you so's..." I enjoyed watching him squirm.

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  38. I see you all have gone to the senile teaching assistant's blog...

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  39. You either misunderstood or didn't inform yourself carefully enough.
    I won't talk on Lubos' place but it is clear that he (and others) don't write about good and evil because this is not a matter of good and evil.
    This is a matter of geostrategy, of realism and of power ratios.
    For instance despite the fact that my sympathy (emotions) go to Ukraine and not to Russia because of my 1968 experience, I find that Crimea going "heim ins Reich" is legitimate, necessary and ultimately stabilizing the whole region.
    Neither good nor evil - a necessary consequence of quasi natural laws of geopolitical gravity.
    It was so obvious for me that I have posted here a few weeks ago my prediction that Crimea WILL go back to Russia, that there will be referendum and that the Crimean will be in majority very happy about that.
    And this is exactly what happened.
    .
    Of course that the Russian population was not really threatened and especially not in Crimea where they represented a crushing majority.
    But by the same token the Sudetendeutsche population was not really threatened in the 1930ies what didn't prevent Munich to happen.
    .
    In this case the Russian government did what every other government would do in its place - use an exceptionnally favourable opportunity window to improve its geostrategical situation at 0 cost.
    Russia wouldn't go to a full fledged war against Ukraine (and whomever would decide to "help" it) JUST because of the Sevastopol naval basis that it has leased and secured for decades anyway.
    The German army called Crimea "the aircarrier in the Black Sea" and this didn't change in 70 years - Russia just took the opportunity to hiss its own flag on the carrier and be applauded by the population.
    .
    In 10 years everybody will forget about this matter anyway like everybody forgot about Kosovo.
    So no it is not about good and evil, it is just about recognising facts, power and geography.

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  40. The problem with Klaus (whom I like for the things he did right: quick establishment of capitalism, clean split of Czechoslovakia, attitude towards AGW) is that he is not independent voice any more and in matters concerning Russia he is biased as hell.

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  41. Ukrainians and Russians....

    Why can't they be more like the Czechs and the Slovaks? A long shared history, some of it under a common roof... finally, a "velvet divorce".

    ... or like the Dutch and the Deutsche? Everything is smiles and harmony between these two. Never is there a harsh word spoken. Really, they are like two peas in a pod.

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  42. The reason I brought up "good and evil" was because Lubos injected a fair amount of black and white in characterizing Ukraine and Russia in his posts on the subject. If he remained all detached and geopolitical, I would have had less of a problem, except he didn't.

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  43. "... the BICEP2 discovery supports the general concept of the multiverse ..."
    "The sum of the energy of matter and of the
    gravitational energy is equal to zero". — Andre Linde
    http://www.mpa-garching.mpg.de/lectures/Biermann_07/LindeLecturesMunich1.pdf

    There is only one way that Linde's theory of chaotic inflation can fail and that is if Newton-Einstein gravitational theory itself fails. However, Milgrom, Kroupa, Pawlowski, and McGaugh have proven that Milgromian dynamics is approximately correct. If the multiverse has neither boundary nor interior, then Linde is correct. If the multiverse has both boundary and interior, then Milgrom is correct.

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  44. In all logic, there would have to be "barrier & interior". The earth isn't flat. Molecules, atoms, bacteria; nothing is essentially flat. In order to have no boundaries or interior, it would have to be "flat".. which doesn't exist at the atomic level. What they will find, when they create the literal model based on the complete correct string theory will be that when zoomed out on a scale of 10mX10^1,000,000.... the Multiverse will resemble a very refined Double Helix.

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  45. Too bad BICEP2 has been determined to be wrong.

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